“no other allegiance.”
I’m open to suggestions on how that works, but nothing I can envision would seem to do it. I don’t think you can say that a person born in America as an American citizen is disqualified just because some other country has some law that says they could be a citizen of another country. Then you are effectively deferring to a foreign country who can run for President in this country. Maybe if someone applies for citizenship of some other country... But if that’s all we’re talking about, then I suppose most people would agree, and we would not have much to argue about.
Many countries use the rule of "Partus Sequitur Patrem" in defining their citizenship. This is such an old rule it has a Latin description. Indeed, it was used by the Romans and the Greeks. This rule was pretty much universal except for England, but by by the time of our Constitution the English were also using it. (I believe I read a snippet from Leo Donofrio's blog pointing out that England only adopted the "jus soli" rule to allow for the Ascension to the throne of a Royal with a foreign father. A bit of skullduggery if you will.)
The point is, the United States recognized a nation's claim on the offspring of its citizens as a very ubiquitous practice, and as such understood that it could lead to a dichotomy of Loyalty between this nation and another. By requiring that a citizen CANNOT be claimed under this universal rule they were insuring that issues of loyalty would be kept to a minimum.
In the 18th Century, it was common practice for Royalty to cross marry and cross rule. The Heads of Europe were all related by blood, and as a result it was not always certain as to where their loyalties lay. The Best way to insure against this kind of foreign intrigue was to be certain that none of our Executives had this kind of relationship with another country. If there could be no claim by blood or by soil, then there could be no foreign claim on our citizen's loyalty.
For most of our nation's existence, MOST of our citizens were born as "natural citizens." Why should we need to seek leaders among those who were not?